 Okay. We are live. Welcome, everyone. Thank you for joining us for today's webinar. My name is Alejandro and I'm going to be today's Love Physics Coordinator. Today we're presenting Testing Einstein with Numerical Relativity, the Precision Frontier and Theories Beyond General Relativity by Leo Stein. Leo is currently an assistant professor in the Department of Physics and Astronomy at the University of Mississippi, also known as Ole Miss. He got his PhD in 2012 at MIT. Then he became a NASA Einstein fellow at Cornell from 2012 to 2015. And then he moved to Caltech as a senior postdoctoral researcher from 2015 to 2018. His research interests are studying and testing general relativity and other theories of gravity from an astrophysical standpoint view. He has worked in many, many interesting projects and this is something he's going to talk about today. But I also want to take this opportunity to invite you to visit his nice website called Due to Symmetry. And then you just put those words together.com. And I believe that's referring one of the most important theorems in physics and mathematics, Nader's theorem. So maybe at the end, Leo might want to talk about his website, which is actually how I saw or knew first about Leo, because he has very nice simulations, information, many, many, like, you should visit that for sure. Remember that you can ask questions over email through our YouTube channel or Twitter. And then at the end, we will give those questions to Leo. So let's start and let us start to over Leo. Thank you very much for joining us today. Thanks for the very nice introduction. Can you hear me? Yes. Okay. And yeah, it's a pleasure to be able to finally make it to one of these webinars. Okay, so hopefully all of the audio visual will work. I'm trying to be in two places at once so that you can see me at the same time as you can see my slides. So I'm going to talk a little bit about testing general relativity. You see that I swapped the subtitle. I'm doing theories beyond GR first, and then talking a little bit about the precision frontier in general, in numerical relativity. So the first half of the talk is going to be about, well, maybe the first three quarters of the talk is going to be about testing theories beyond general relativity. And then the very last part is what we need to do in numerical relativity so that we actually have the precision to say that we are testing general relativity. Can you see my mouse, by the way, if I point to things on the screen with us? Yes. Okay, great. Okay, so here's a little status update of what's going on in observational astrophysics in the last few years. So as of 2015, we've added a lot of dots to this figure of the stellar graveyard. So all of these blue dots at the top are binary black holes that have merged together that were observed by the LIGO and Virgo observatories. They emit gravitational waves, and that's how LIGO and Virgo are able to detect them. And hopefully that you're a little bit familiar with how LIGO works, I'm not going to spend any time talking about it. I know that Nicholas Yunus gave a talk back in November, so I think he talked a little bit about how LIGO works. And then there's also this one pair of neutron stars that merged in 2017. So before 2015, we didn't have any black holes known to humans above this mass. We didn't have any of the blue dots, and we didn't have these two orange dots. We just had the yellow dots and the purple dots. So we're already learning a huge amount about black holes and neutron stars that we didn't know before through this new observational channel of gravitational waves. So here's a little family portrait of the 10 black holes that have been observed so far. This is a bunch of simulations that were made by Teresita Ramirez and Jeffrey Loveless, who are in the Simulating Extreme Space Times collaboration that I'm a member of. You can find this on YouTube too. So there are 10 of these that we've observed. You see they all have different mass ratios, and down at the bottom there are these gravitational waveforms wiggling from left to right, showing you our best reconstruction of what it was that made LIGO go ping. So here's a little more numerical way to look at it. Instead of these simulations, this is actually what LIGO saw. They saw some signals in time and frequency space. So time is going from left to right in each of these panels, and frequency is going from bottom to top. So this is the characteristic pattern that two merging black holes make in gravitational waves. As they spiral around each other losing energy in gravitational waves, they're emitting some frequency that's roughly twice their orbital frequency, and then as they're losing energy to gravitational waves, that means that they fall even deeper into each other's gravitational potentials so they get closer and that makes them go faster and that makes the frequency go up. So right when they're about to merge, they make a gravitational wave that kind of goes like, and we've seen 10 of these so far. So this is the family portrait. So we've got gravitational wave detections. We know that they're here to stay. We're going with the advanced LIGO detectors and Virgo detector are being upgraded right now. The Japanese detector Kagura is going to come online soon to join them. There's a new gravitational wave detector being built in India as a collaboration with LIGO. So we're going to have new science coming out of gravitational wave detectors for the next, you know, hopefully many decades. And we know some of the science that we're going to get out of them. And there's a bunch of science that we hope to be able to get out of them. So here's what we know that we're definitely going to learn. Once advanced LIGO is up to design sensitivity, they should be observing black holes merging with each other roughly once every week. So there are going to be enough observations that you'll be able to say something about the populations of binary black holes out in the universes. What are their masses? How are they spinning? Do we think that these binaries were formed in globular clusters or out in the field in galaxies? And what kind of systems formed them and how did they evolve and those types of things? Then there's a lot of things that we wanted to learn about neutron stars. So a few years ago, when there was the first version of this slide, I was saying we're going to find out whether or not neutron stars are the central engine that powers gamma ray bursts. But now I think that after the event GW-1708-17, the one neutron star, neutron star merger that was seen, we know that neutron stars power gamma ray bursts. And we think we know how much of the heavy elements are formed in neutron star mergers. And we want to know about the properties of the dense nuclear material that neutron stars are made out of. And that we can learn by how squishy are neutron stars, how much do they deform when they get close to each other. So that shows up in the gravitational waves and hopefully we'll be able to learn about nuclear matter at densities that are higher than anything that we can probe on Earth. So these are things that I think that we're going to learn in the next decade. And I'm pretty confident that we're going to do it. So I'm not losing sleep over whether or not we're going to learn these things. The thing that I really care about is testing general relativity. So I don't know whether or not we'll be able to detect if gravity is any different from general relativity or maybe general relativity is the correct description of gravity in the universe. So this is the thing that I worry about. So let me tell you why I worry about it. There are a few reasons. So these are Einstein's field equations. This had to be the first equation in the talk. So on the left hand side, we have a curvature in space time. This is describing some geometry. And on the right hand side, we have some sources of matter creating energy and momentum flow. And I put a little hat on top of the T, because all of the matter that we understand is made out of quantum fields. So the thing on the right hand side is an operator. And the thing on the left hand side is classical. So this doesn't make sense. And if you try to make sense of it, you run into all sorts of paradoxes. So if you try to make GR into a quantum theory, you find out that it's not renormalizable. And if you try to do something slightly different, general relativity is classical, but still treat all of the fields as quantum mechanical, then the calculations still tell you that you need new physics. So this, the most obvious place where you see that there must be new physics is in the black hole information paradox. So what happens is when you combine quantum fields on a classical black hole space time, you find that black holes actually evaporate. This was Stephen Hawking's most important contribution to physics. So that doesn't sound so crazy. I mean, well, it sounds a little crazy, but what's the problem with black holes evaporating? Well, as they evaporate, they get smaller and smaller, so curvatures get higher and higher. So all you have to do is have a black hole and wait, be very patient. And eventually, the black hole will make such strong curvatures that you need a quantum theory of gravity because the curvature will approach the plank scale, so you don't know how to describe it. Okay, but maybe that's not as big of a problem. Maybe you think there is some quantum theory. So why else should I care about black hole evaporation? Well, the other problem that we learned by studying Hawking radiation is that the Hawking radiation itself carries information away. And it's telling us that the information that of the stuff that fell into the black hole is being destroyed, and we're not getting a back. When you turn that into math, that says that probabilities stopped adding up to 100% or if you want to be more mathematically precise, then the theory has stopped being unitary. And that's a real problem because we don't know how to do calculations in quantum mechanics if things are not unitary. So these are the reasons that I feel compelled to study how general relativity might be corrected by some maybe quantum theory of gravity, but maybe there's just some new gravitational physics that we don't understand. So there are two ways to go about studying it. The first approach is to study quantum theory, to study string theory, to study loop quantum gravity, or some other approaches to fundamental theories and to try to find a quantum theory of gravity that resolves these paradoxes. I'm not going to talk about that because I don't know how to do that. So the other approach is the empirical approach. So sometimes in the history of physics, theory has made predictions about experiments and then we went and looked for it and found it to be true. But other times, people just went and did experiments and found something that theories did not describe. So this is the other approach. So the idea is just to go make as precise observations as we can and see whether or not they all agree with general relativity. And maybe nature will tell us how we're supposed to change the theory to make it describe nature better. So so far, we have a lot of precision tests in the weak field and I'll tell you in a slide what is meant by the weak field. But there are a lot of theories that are like general relativity in the weak field. So the place where it hasn't been tested very precisely is in the strong field. So that's when two black holes are merging together. Curvatures get very large and the theory is very nonlinear. Nonlinearities are very important and it's dynamical. So the space time is changing on times that are very small compared to the light travel time across the system. So dynamics are important, which is not true in any of the other regimes where general relativity has been tested with high precision so far. So here's a nice figure from Tessa Baker and collaborators. That's kind of the phase space of all of the all of the gravitational experiments that have been performed so far to date. So on the horizontal axis, there's something like the strength of the gravitational potential, the Newtonian potential. And when we get up to one, that's where a black hole forms. And on the vertical axis is the strength of curvature. So going up is more curvature. So what you can see is that there's stuff in the solar system down here in the middle of the figure. I think I'm not presenting to everyone. Like we are seeing your screen and also like your little video. And so if you look at the slides, then you'll see that down here are things in the solar system. And here there are objects around the galactic center around our supermassive black hole. And all the way at the top right are neutron stars and black holes merging with each other that's seen by LIGO. So over here, these are tests that might be done by the event horizon telescope. So they're overdue for an observation that they made based on data that they took two years ago. So pay attention and see if they have any observations announced soon from the event horizon telescope. But I'm just going to be talking about stuff up here about black holes merging. So up here dynamics are important and curvature is important. And we don't have precise tests of general relativity up here yet. So here's a big picture. Here's where GR has been tested and hasn't been tested so far to date. Before advanced LIGO, we had precision tests in the weak field. Now with advanced LIGO, we have access to the strong field, but we don't yet have a precision test because the detector is not up to design sensitivity yet or we haven't just gotten lucky enough yet with very loud events. So so far we've been able to say things like the gravitational waves are consistent with the predictions from general relativity at something like the 3% level in some model independent way. But in the future, the detector will get better and we'll get more observations. So we want to make precision tests of general relativity. And so I'm just going to focus on binary black holes so we don't have to worry about the problem of whether or not nuclear matter is is tricky. And there's something that we don't understand about nuclear matter in neutron stars in black hole binaries. The only thing you have is space time. So there's nothing else to worry about modeling. Okay, so the question is, how do you actually do precision tests of general relativity in the strong field? So here there are two, two paths that you can take. One theory specific, you can choose your favorite theory or your least favorite theory, the one that you want to destroy and do a lot of calculations and do all of the modeling and compare against observations. And the other one is theory agnostic. And in the theory agnostic point of view, you try to to parameterize what kinds of models might come out of all of the different theories. So here's some examples of this parameterized post Newtonian and parameterized post Einsteinian that I'll talk about briefly. So first of all, when you have compact objects that are separated from each other by a large distance, you can do this analytical trick where you treat them as point particles, but these point particles have some more properties. And these properties are their multiple moments. So on the left, you see the multiple moments of the earth that's been measured by satellites doing geodesy. And the scale has been blown up a bit. So you can see that the earth's gravitational field is actually bumpy. So in other theories, you can also go and compute what are the bumps, what are the multiple moments of black holes. And this has been done for a number of different theories, some of which I list here, and you can ask me about them if you want more details. So this is the theory specific approach, right? All of these things that I've listed here are you choose a theory, and you take the equations, you try to solve them with neutron stars or black holes, and find out what the multiple moments are. And then you can do post-Newtonian calculations. I'll skip that. The other approach is to just parameterize the multiple moments. So take what we've learned from all of these theory specific calculations, and now that we have some experience say, okay, so if somebody gives me a new theory, it's probably going to look like this. So I can parameterize what the multiple moments are going to look like. And so this is just a figure from a paper with a kentiyagi, where again, we have on the horizontal axis the compactness of some small object. And on the vertical axis, the strength of curvature. And here are some solar system bodies. And what you're seeing is the bounds that you would be able to place on some theory's length scale. So every way that you change gravity, you have to introduce a new length scale. So that's the vertical axis on the right side of the figure. So for these two different theories that I put on here, Einstein-Delta and Gauss-Benet and dynamical-Chern-Simons, if you measure bodies that are more and more compact, you can bound this length to being shorter and shorter, which is saying that you're constrain the theory more and more. So you want to push the constraints up. So what you can see is that with pulsar timing, you can make constraints at one level at maybe the 10 to the 4 kilometer level. But once you go to gravitational waves up here in the top right, you can push things down to the 10 to the 100 kilometers or maybe even 1 kilometer level. Okay. So that was parametrized post-Newtonian. And kind of in a similar spirit, we can do something called parametrized post-Einstein. So this is something that Nico has worked a lot on. So the idea is that we take the gravitational waves that are predicted by general relativity. And instead of just taking those predictions, we multiply them by some correction. We do this in the frequency domain because that's where the results of the post-Newtonian calculations come out most easily. So you multiply by some amplitude correction, and you add a phase correction. And you can just make up these numbers, alpha, A, beta, and B. You can also go back to your favorite theories and calculate what are alpha, A, beta, and B in that theory. But this form works for a lot of different theories. And we know it works for a lot of different theories because we have experience doing theory-specific calculations. So this kind of calculation was part of LIGO's first testing GR paper. And in their most recent testing GR paper, which was on the archive a week or so ago, they have updates on this. So each of these columns are one of these parameters being constrained. How far away can it be from zero? So zero would be general relativity. So all of these parameterizations that I told you about, they make sense in the weak field. But let me go back to this. So this calculation makes sense when frequency is small. That's where we have analytical control over it. U is proportional to the frequency. U cubed is proportional to the frequency. The problem is that in all of the black hole mergers that we're seeing, the frequency is not small. Or really, the velocity of the two bodies when they're merging is approaching the speed of light. And that's the expansion parameter in the post-Newtonian calculations. So that's not small. So we don't know if this calculation is under control in the merger regime. So we want a better parameterization, but we don't have any examples to work from. So what do we do? The reason that we don't have any examples to work from is because of this thing called the initial value problem. So let me just skip this and tell you what the initial value problem is. So when we do numerical relativity, we have to contend the facts that general relativity has a certain mathematical structure. It's nonlinear. It's second order partial differential equations. There are 10 metric functions. There are three coordinates, a one-time coordinate. What you need to do is you need to cut up time so that you can put it on the computer. So if you're just thinking about geometry, then the equations of general relativity look very simple and beautiful. It's just GAB equals 8 pi TAB. But if you want to put it on the computer, you have to do this. Let me make sure that, sorry. So Alejandro, can you make the other copy of me be presenting to everyone? Like the PDF? Yeah. Okay. Done. Great. Thank you. Okay. So right. So these are some very complicated equations and it all comes from the fact that you need to chop up space time into space and time. So you have to kind of break general relativity in a certain way. People started working on this in the 1960s and they only started merging black holes. Oh, I guess this is 14 years in 2005. So to be able to do evolutions, you need to know about this thing called the initial boundary value problem, which means if you give me data at some time, how do you get the data at some future time? That's maybe one second later. And we know how to answer that question with certain types of partial differential equations. But general relativity is hard because it has gauge freedom. So there's no unique way to describe the data on any time slice. There's always freedom. So you need to fix that gauge freedom. But even if you fix it, you still have to satisfy certain constraints so the data are not totally free. So this is kind of technical and you can ask me about it later if you're interested. But people finally solved this problem thanks to Fran's Pretorius in 2005. So now we can do simulations like this. This is also on YouTube. If it's too choppy over the presentation, you can find this video. So you can see a slice through spacetime in the top panel. And on that slice, we're showing colors that show you how strong is curvature, one certain part of curvature. And also the direction that spacetime wants to flow. And you see we have these two black holes and spacetime kind of wants to flow down into the black holes. And on the bottom, what we get out of it is the gravitational waves. Okay, so we know how to solve this problem for general relativity. It has a good initial and boundary value problem. But it took us from the 1960s until 2005 to be able to do this on the computers. So every other gravity theory that you can think of is going to be at least this difficult. So here's an example of a gravity theory where we do know how to simulate the theory in full on the computer. It's really just general relativity with an extra scalar field. So let me skip over this. But there's a whole lot of other types of theories, and there's a whole lot of problems that they can have. So they can have higher derivatives. And that means that people don't really know what to do with the equations. They might be unstable. They might create these things called ghost modes, which again, if you're interested in what these are, then you can ask me about it later. They might stop being quasi linear, which from my point of view is a very bad problem. And I don't know how to solve that one. So I'm not going to deal with any theories that are not quasi linear. But I will tell you one possible solution. And this is for the types of theories that I've been working on. So the idea is to treat treat some theories as general relativity, plus some small correction. So we kind of have an idea that that has to be true already, because all of the observations that we've seen are consistent with general relativity. So we can't be too far off. So we do this in the standard model already. This is called effective field theory. And the point of view is that general relativity sits inside of some bigger theory that we don't know about yet. And what we can do is use the tools of effective field theory to go from the bottom up. So kind of bootstrap ourselves from general relativity to be able to treat a theory that is close to general relativity. So the main tool that we use is perturbation theory. And as long as we're close enough to general relativity, and there's a new energy scale of this new theory, as long as we're below that energy scale, we can use this ratio of energy scales as a small parameter to control the calculations. And when you do that, the structure of general relativity is really what determines how these equations get put on the computer. So here's an example. I'm not going to give you all of the details, because we don't have enough time for that. But this is a specific theory called dynamical churned simons, where this part here, the R, this is the action, if you know what the action is, is just the part coming from general relativity. And now we've added something that's like an axion that's non minimally coupled to curvature. And the equations of motion are here are Einstein's equations. But now I've added this extra term with the epsilon in front, and epsilon is supposed to be small. And this term has third derivatives of the metric. So if you gave this to a mathematician, they would say there are third derivatives of the metric. There's only one person who knows how to deal with that type of partial differential equation. So I don't know what to do. But if you use perturbation theory, then it's this term that controls the structure. So it's the Einstein tensor that controls the calculation. So very briefly, we know lots of things about black holes in dynamical churned simons. So here are some movies of solving for the structure of a black hole in dynamical churned simons. But this is without using full numerical relativity. There's no time in this simulation. Every frame of the simulation is a different value of the spin of the black hole. But there's no evolution in time. So I don't have to worry about the problems with well-posedness here. So we know that black holes have certain structures in churned simons. That's because we did lots of analytical calculations. We even know what happens when you put together two black holes in a binary in churned simons. So they have kind of a dipole moment, kind of like magnets. So as they go around each other, these dipole moments will also process, just like the spins in general relativity process. And that will also lead to the extra axion field getting radiated out. So this was all known from analytical calculations. But nobody had ever tried to do the merger because we didn't have a numerical relativity scheme for this other theory that's not general relativity. So as I mentioned earlier, if you expand in powers of this small number epsilon, which depends on this new length scale or new energy scale, then it's really general relativity that controls how you put the theory on the computer. So you can ask me about this in more detail if you want to know how this goes. But the punchline is that when the background theory, general relativity, is well posed, then the corrections to it, meaning the corrections from this extra tensor that we added to the field equations, that correction is also well posed, which means that we can actually do simulations in a theory that's not general relativity. So these were the first numerical simulations of two black holes merging in this theory that's not general relativity. And there's a lot of stuff going on in this visualization. But the important thing to take away is that there are two black holes going around each other, they're making radiation in this axion field that you saw going out very far away. And then they merge and this makes a burst of radiation. So this is similar to the gravitational radiation that we have in general relativity. But this is a field that's new in this theory, general relativity does not have this axion field. Okay, so we can actually do this. So we can do this more quantitatively besides just making movies, we can show what these waveforms look like if we had an axion field detector. We don't have an axion field detector, we have gravitational wave detectors. So this is what happens at the first order in perturbation theory. So if we want to know what gravitational waves we get, we need to go to second order in perturbation theory, which is what we're working on now. Still, even with the first order perturbation theory results, we're able to make estimates that informed by those simulations, we can say once we get the second order simulations done, then we think when we plug in all the numbers that LIGO will be able to constrain this new parameter, this small number L over GM, or L itself is a length to being, you know, if LIGO does not see any difference from general relativity, then we know that this new length scale L has to be less than something like 10 kilometers. So this is a seven order of magnitude improvement over the bounds that you can place in the solar system. So this is the real key of going to the strong field, highly dynamical, strong curvature regime. You're getting seven orders of magnitude improvement. Okay, so as I told you, right now we're working on the second order perturbation theory results. So this is work that I'm doing with Masha Okunkova, who is a grad student at Caltech. This is her self portrait. So we're starting with the easy thing first. The easy thing is, instead of doing a full, in spiral and merger and ringdown simulation, we're just doing two black holes merging straight into each other, just a head on collision. So this is not something that happens in nature, as far as we know, but it is a good way to probe what happens after two black holes merge, because when you merge these head on, the only gravitational waves that you get is from the merger itself instead of from the in spiral. And you get a black hole that is highly distorted. And then it rings down after the merger, which you saw in these plots here. So I think I can zoom in. Yeah. So this part of the waveform from our simulation is called the ringdown. So a new black hole has formed from the merger of two smaller black holes. And it's very distorted. But there are these theorems in general relativity that tell us that a stationary isolated black hole in vacuum is unique. So it has to be eventually described by what's known as the Kerr solution. And the way that it becomes the Kerr solution is by shaking off all of these deformations in a bunch of characteristic quasi normal modes. So they're like normal modes, except they're quasi normal because they're they're damped and they don't form a complete basis. Sorry, this is the wrong size. Okay. So those those frequencies are very characteristic of what is the final mass and spin of the black hole. And there's an infinite number of these frequencies. So here's an example from general relativity of a of a head on merger. So this is two black holes with their spins pointed at each other. And they crash directly into each other. And and here are just three of the infinite number of modes that come out of this ring down. And they're labeled by some numbers 204060. So the horizontal axis is the time of the simulation. And the vertical axis is the strength of something called psi four, which is like two time derivatives of the metric. And this is on a log scale. So what you see is some envelope that's going down linearly on a log scale. So it's exponentially damped. And there's a different frequency for each of these modes. And that frequency depends on the mass and spin of the black hole. Now the mass and spin are only two numbers. So in general relativity, the infinite number of modes are all determined by just two numbers. But when you change the theory, they might be in different relationships with with each other. So the point is, if you change all of these frequencies, because you change the theory, there's probably not any black hole in general relativity that has those same ring down frequencies. So this would be a smoking gun of whether or not the theory is general relativity or if it has been corrected. So I'm not going to show you the results yet because Masha and I are still working on finishing this paper. But I'll show you a cartoon of what we're finding so far. So these are two example ring down waves. And they have different amplitudes and well they have different decay times, but they actually have the same frequencies. So it's only the envelope that I changed here. So what we're finding is that the dominant correction to these ring down modes is that the decay time changes but not the frequency. This may be specific to the head-on merger case. We don't know yet, but we should have this paper out in a month or two. So watch out for that paper. This will be a hot results. You heard it here first, but I'm not giving you the actual result because it's Masha's results to present. But you might have expected this to be something plausible because now there's another field, the axion field that we added to the theory that takes away energy faster than energy leaves the system in general relativity. So that's kind of a plausible explanation for why the decay time is faster when you change the theory than in general relativity. Okay. So there's a huge amount of work that we still have to do. So we're still working on the second order thing. Once we get all of these things working, we have to do all of the same stuff that we do for general relativity. And that means run simulations all over parameter space and then build waveform models. There's something called surrogates that I'll talk about in a little bit. And I'll tell you why we have to build these surrogates. And then we have to study whether or not the waveforms that we computed from a theory that's not general relativity can ever be degenerate with those from general relativity, which is to say if there is a change to the theory, but the calculation or but the observation, the prediction is the same as the GR prediction, then we would never be able to test it. So if it's degenerate, then we just can't test this theory. Now I just gave you an argument here for it not being degenerate, but really we need to do a thorough investigation with a full waveform model overall of parameter space to know whether or not there are regions of degeneracy. Okay. So now let's step back and say it's a few years from now and we've been able to compute all of these simulations, build waveform models, etc. And LIGO is at higher sensitivity and they're getting better observations. And somehow we observe a gravitational wave that does not agree with a prediction from general relativity. What does that mean? So there are a few possibilities. One possibility is that general relativity is wrong and there are new, there's new gravitational physics that we have to go learn about. That's the most exciting thing for physicists because it, you know, there's new stuff to learn about nature that we don't know yet. But there are some more mundane reasons that this could happen. One is that there are types of noise in the detector that we didn't understand yet. So the experimentalists and the people who work on detector characterization have to work very hard to get rid of all of those noise sources. The other possibility is that our waveform models are not good enough. So when we do numerical relativity, we have to pick a certain number of grid points to be in the simulation. We don't have infinite resources. So that means that we're always making some error. We're not really getting the full waveform, we're just getting an approximation to the waveform. So okay, so it's expensive to do these simulations. Depending on the resolution, it could take maybe a weekend or it could take a week or it could take a month. So we can't do enough numerical relativity simulations that we would be able to use them directly for parameter estimation for LIGO data. Instead what we do is we run simulations all over parameter space and then we build some analytic models with a bunch of free parameters and these go by names like SEO, BNR or IMR, Phenom. And if you want to know what these are, then please ask me. But they have free parameters and they are calibrated to simulations that come out of numerical relativity. And the way that we calibrate them is by making sure that they have as good of a match as possible. So if you know something about gravitational wave data analysis then this is familiar to you. But basically we're kind of taking two different waveforms and looking at how similar they are in the space of all possible waveforms. And we normalize by the waveforms themselves so that the total amplitude doesn't matter but just the shape of the waveform. So that's the match. So if two waveforms are identical then they have a match of one and the mismatch is one minus the match. So the idea is you build your analytical model, your calibrated model like SEO, BNR and you change the parameters, the free parameters in there until the mismatch between your model and numerical relativity is as small as possible. So this has some error as well. So the numerical relativity simulations have some error and these analytic models that we build also have errors. So here's an example of these models and the errors. So the black curve which you can see through the dashed line is the numerical relativity result of some kind of complicated binary. It has some mass ratio of 8 and there are spins that are pointing in kind of arbitrary directions and that makes the orbital plane of the binary process which is what leads to all of these little modulations in the waveform. So there's the orbital frequency but then there's also some procession. So the blue curve is from the calibration SEO, BNR, V4, version 4. And what you can see is that at early times it matches very well but eventually once you get to the merger regime there's a bit of phase difference between the SEO, BNR waveform and the numerical relativity waveform and there's another curve on here that I'll talk about in a second. But this is the kind of error that's a problem. If you're using SEO, BNR to test general relativity then there are discrepancies between what we think is the truth and our model that could hide. So we wouldn't be able to tell if there's a difference between our model and nature or between nature and our best theory. Okay so here's how to make this quantitative. So the horizontal axis is the mismatch. This is from a study done by VJ Varma who is another one of my colleagues at Caltech. It's another grad student and collaborators. And the vertical axis is how many simulations have, this is a histogram, how many simulations have this level of mismatch. So remember you want mismatch to be zero. That would mean that you had a perfect model. So you want things to be on the left. So SEO, BNR, V4 has this distribution of mismatches here that's peaked around let's say 2 times 10 to the minus 3. What that means is if you're using SEO, BNR, V4 to do data analysis with LIGO, you would never be able to make a test of general relativity that is more precise than the tenth of the percent level. You can't do better than how well the model is agreeing with numerical relativity. So you can't do precision tests unless you improve the model. So the black histogram on here is the errors that we've assessed from the numerical relativity simulations themselves. So how do we assess them? We take the same simulation at one resolution and then we increase the resolution and do the same simulation again. So now we have the same binary but one is with higher precision. So the higher the resolution, the closer it should be to the truth and then we compare these two different resolutions. So this is really the mismatch between our second highest resolution and our highest resolution. So it's not really the errors in the highest resolution. It's a conservative estimate of the errors because they're really the errors in the second highest resolution. And then there's this other thing on here called a surrogate model and our hype sur where the mismatches are down here peaking around 10 to the minus, well a few times 10 to the minus six. So what this is telling us is that these things called surrogate models are agreeing more with the highest resolution simulations than the medium resolution simulations are agreeing with the highest resolution simulations. Or in other words, these surrogate models are basically recovering everything in the numerical relativity simulations. They're doing better than the medium resolution simulations. Okay, so what are these surrogate models? So the idea is as follows, here's a movie that was made by VJ Varma. So at every point in parameter space, here it's Q, which is running across this axis that's coming towards us. We do a simulation. So we have a whole bunch of different simulations at different points in parameter space. And now we fit some, let's say, amplitudes to these and phases to these. And then we choose a bunch of common times, common amongst the different simulations, and we build interpolating functions. So these orange curves are interpolating functions over the different parameters, which allows us, let me just pause this here. So once we have the orange functions, that allows us to pick a new parameter that is between the simulations that we've run so far. So we haven't run a simulation yet at this value of mass ratio and get the waveform out at that new parameter. And after you've built the model, it can take something like just, you know, 30 or 50 milliseconds to get the full waveform instead of a week or a month on a supercomputer. So that's a surrogate waveform. And these surrogates are fast enough that I can do this. Oh, okay. Sorry. This is a movie. So this is not difficult for my computer to do. It's just showing you a movie. But instead of doing a full numerical relativity simulation, you can just post process the surrogate to make movies of what happens in these simulations. All right. But let me show you how fast these models are. So I have to Oh, sorry, sorry, sorry. Exit out of this and run some Python code. So this is Python. This isn't even something compiled. Okay. So it's a little difficult because I'm also doing video sharing on my computer. But this waveform itself only takes 50 milliseconds to compute. And you can see we can change the point of view and the waveform is changing in real time. You can install this Python package on your computer with one command. So this is all based on the surrogate models. It's fast enough that you can basically simulate a black hole binary on your laptop in Python with 50 milliseconds of computation time. Okay. Now my computer is going too slow. So let me get back to this. Okay. So the point is, I think I already said this, the surrogate model is essentially capturing all of the information in the highest resolution simulations, which means that it's doing better than our medium resolution, which is the black histogram. Okay. So let me just tell you about some, some, okay, let me go back to this and say ideally we want to take the black curve and the surrogate curve and push them as far to the left as possible. We need to make our simulations good enough that we don't have to worry about the simulations. All we have to worry about is our modeling uncertainty. So if we want to do precision tests at the, you know, at the sixth decimal place, then we need all of these mismatches to be less than 10 to the minus six. So some of the, some of the things that we do in the numerical relativity simulations right now are imperfect and we need to improve them to get the simulations down to that 10 to the minus four, 10 to the minus six level. So, so one of the problems is that you can only simulate a finite volume of space time because you only have, you know, a finite number of points that you can plop down on the computer on the grid. So after the gravitational wave leaves the box that you're simulating, it still has to climb out of the gravitational potential that's created by this binary as it's going around. And that might sound simple, but there are actually nonlinear effects for the waveform propagating out through this potential. So for example, the waveform will redshift a tiny bit as it propagates out because, you know, deeper in the potential, it's just like, well, it literally is a gravitational redshift experiment, right? The gravitational waves climb out of the potential. So they have to expend some energy. So they end up being at a lower frequency. So, so that's an effect that most numerical relativity codes don't take into account, because the usual way of getting the waveforms out is to extract the waveforms at a whole bunch of different radii inside your domain. And you know that these, these gravitational waves just like electric fields and magnetic fields, they fall off like one over the distance away from the source. So you just do a fit to the part of the wave that's, that's decaying like one over the distance. And once you got that fit, you can, you can then, you can, you can estimate how large the gravitational wave would be at any distance away because you know what the one over r piece is. What that's neglecting is these nonlinear effects from climbing out of the potential. So the right way to do this, quote unquote, right way to do this is, so this is a space time diagram that, that Masha made. The right way to do this is to take our simulation volume, which is this orange region. This is a world tube. Time is going up. Our world tube has some finite size to it because we can only put down a finite number of points. So we take the gravitational waves at the edge of our world tube, and then we want to propagate them along light rays going from the boundary of our simulation out to the boundary of space time. And this is something that you can actually do mathematically. You can actually squeeze down along these light rays to be a finite number of points. This is called compactification. And then if you evolve the gravitational waves along these null rays, along the light rays, that's called characteristic extraction. So this, this whole scheme coupled together would be called Cauchy characteristic extraction. And there's even one step further called Cauchy characteristic matching, which I will, well, you can ask me about that if you want. So, so this is something that we're working on in the collaboration right now, making this be, we have code that does this, but it's not in production. But the next generation of waveforms that we'll be producing will have this effect taken into account. So there, there, there shouldn't be any more discrepancy because of the waveforms being evaluated at a finite distance. But we're actually, but we want them at, you know, arbitrarily large distance. Okay, so that's, that's one thing that we're working on improving. Here's the second thing that we're working on improving. So here is a view of parameter space from an upcoming paper that I'm an author on. So the parameter space for these black hole binaries has total mass, which we don't have to worry about for black hole, black hole numerical simulations because it can be scaled out. Then it has mass ratio between the two bodies. And each of the two black holes has a spin vector. So that's three numbers for each black hole. So that's seven numbers. Right now, we're only doing quasi circular binaries. So we don't have eccentricity. And we don't have an argument of Perry center. So that's two more numbers that we could be that's two more dimensions to parameter space that we could be simulating that we're not simulating right now. But even in the parameter space that we are simulating, here's a way of viewing where we have simulations and where we don't have simulations. So you can see a whole bunch of projections two dimensional and one dimensional projections of this seven dimensional parameter space. What you can see is that it's very difficult for us to go up to high mass ratio. And I can explain why that is. And it's also very difficult for us to go up to high spin. So most of our simulations have a spin that's less than 80% of maximal, except for a very small number of simulations up here that go up to maximal. And most of our simulations have a mass ratio that's less than one to four. And so this is all room for improvement. Because it's possible that LIGO will be detecting some simulation, some, some gravitational wave merger event with a mass ratio of five with a spin that's here and this spin is here. And we don't have a simulation out there yet. So our surrogate waveform model isn't good enough there at the SEO B and R might be good there. But we don't really have a ground truth simulation from numerical relativity to compare against to know whether or not it's good enough there or not. Okay. So this is all room for improvement. And there's also the fact that we don't do eccentric simulations yet. So there are a whole two other dimensions to parameter space that we haven't even touched. All right. So that is basically everything that I wanted to tell you. Here are some of the things that I want you to take away from this. So we've been working on the first binary black hole mergers in this theory called dynamical churn Simons, which is also the first higher curvature theory, higher derivative theory past general relativity that anybody has simulated with full, you know, simulate the full merger problem in these theories. And using these simulations, we can make estimates of how well LIGO will be able to constrain those theories if there is no deviation from general relativity. So right now we're working on order epsilon squared and stay tuned for a paper in a few months from Masha Okunkova and myself and others. So that's just on the beyond the GR stuff. And here's the big picture. So we want to test general relativity because we know that there has to be something else going on in nature. We have all these paradoxes information being lost in black holes, Hawking evaporation, stuff like that. So one way is to go looking for new physics just observationally, or you can go and model different theories beyond general relativity. And to to either way to model to test general relativity, we need to make sure that what we're seeing when we compare observations with numerical relativity data from just plain old general relativity has to be sufficiently precise that we're not seeing our simulations noise. So we need to make sure that our simulations are precise enough that we can test general relativity. So there's a lot of work to do there as well. There's a better surrogate modeling. I told you about Cauchy characteristic extraction. There's going to higher mass ratios, higher spins, higher eccentricities. There's a lot of work to do. So it's going to be an exciting few years in this field. So I would be happy to take any questions. So thank you for having me. Thank you very much. Let me see if I stop here presenting you for this fantastic webinar. Okay, I'm back. Good. Thank you very much, Leo. It was a very nice and it looks like you presented like very new material. So it's very excited. And it's very nice that also to see those simulations. We have at least two questions here from the YouTube channel that I post them. Maybe you can also read them. So one is like, you said that the formula in the Palestinian framework can be used with various theories. Does it work for even contradictory ones? That's a good question. And then she commented that actually what the person means by contradictory is theories that aren't compatible at all. Right. Okay. Yeah. Okay. So that's a very good question. Different people have different, yeah, that's a very good clarification to make because different people will have different levels of what is, you know, an inconsistent theory. So there are some theories where they're internally self consistent, like Brand's Dickie theory. But they might have been very strongly constrained by observation so far. So we might say that it doesn't agree with observations. So for example, in Brand's Dickie theory, you can do parameterized post Einstein. That's a theory that contradicts, well, having large Brand's Dickie parameter would contradict what we observe with nature, but there's no internal inconsistency in the theory. So if there was an internal inconsistency with the theory, even then, there's still some clarification that needs to be made about what is contradictory or not. Some people might say, if your theory has higher derivatives, you think it's unstable, something like that, then you should just not worry about the theory at all. Chernzymons, dynamical Chernzymons is actually one of those theories. If you treat it as like a fundamental theory, it probably does not have a well posed initial value problem. In fact, we think that it has instabilities. But these instabilities, actually, if you can share, oh, you're still sharing my slides. Good. Okay. So let me go back to this slide. All right. So we know that there is an instability in dynamical Chernzymons, but this instability is that momenta, a sufficiently high momenta, that it is above what's called the cutoff of the theory. So I think to answer your question, yes, you can do it for theories that from one point of view are contradictory. Some people would look at dynamical Chernzymons and say it is contradictory. But I would take the point of view that this is not a full theory. This is just an effective field theory. And from the effective field theory point of view, it's okay that it has some problem of very high energies. At low energies, I can still do parameterized post-Einstein, and there it works. Okay. And then the second question was what software and or programming language do you use for the simulations? Okay. Good question. So the spec code, spec is the spectral Einstein code is mostly written in C++. And the next generation numerical relativity code called specter, which doesn't stand for anything. So specter is being developed open source. You can find it on GitHub. So if you wanted to participate, you could fork it and start running tests and adding code yourself. That's also written in C++. And we use a bunch of frameworks because these codes have to run in parallel on supercomputers, so they have to pass messages back and forth between the different CPUs and the different nodes that they're on. So spec uses MPI and specter is using something fancier called charm++. And I can tell you more about these things if you want to know. But hopefully that answers your question. Okay. And we have another two questions from Shantanu Desay. I'm pretty sure I pronounced that wrong. Sorry. Are there any cosmological implication of dynamical insurance, Simon's gravity? Okay. Hi, Shantanu. Thanks. That's a good question. Yes and no. So there are calculations of dynamical insurance, Simon's on cosmological scales. I think they might be hard to find, but I can dig them up if you're interested. But there's a simple reason why dynamical insurance kind of hides from cosmology. And that's because all of the correction in dynamical insurance, Simon's is due to this parody odd Cheren Pontriagin term. So if your background is spherically symmetric, like it is in Friedman, Robertson, Walker, Lemaitre, then that background value of the Pontriagin density is zero. So when you start doing cosmological perturbation theory, you don't get any corrections until you hit the second order in perturbation theory. So what can happen, though, is that you can have the axion like field in dynamical Cheren, Simon's can evolve. It can have the same symmetries as FLRW. It can be spatially homogenous and isotropic, but then it will evolve in time. And the derivative of that axion field actually controls a very simple to understand physical effect, which is going back to parity violation, that left helicity and right helicity gravitational waves will now propagate differently because of this axion that they're coupled to. So the main thing it does is I don't remember whether it's left or right, but one of those modes of gravitational waves will be damped faster and the other one will be amplified. Now, that is a potential way to constrain this theory, but what hasn't been done yet is to kind of put that calculation in the effective field theory framework and say, let's say it was the same theory that I'm constraining both from black hole binaries and from cosmology. So you need the calculation to make sense in both calculations, which means that the black hole calculation will constrain the new length that you're adding to be extremely tiny, where order of kilometers, which is tiny compared to cosmological distances. So once you do that, then I expect, but again, I'm not positive because I still have to do this calculation with Bob McNeese. I expect that when you take the constraints from black hole binaries and you apply them to the same theory, but on cosmological scales, then it probably just makes all of the effects too tiny to see. Sorry, sorry, I was talking without my guard. Thank you. Okay. Roberto, do you have a question before we leave? We are running out of time. I have a couple of questions, but more kind of naive in the sense because I'm not so experienced in the area. But first of all, I want to thank you because it was super interesting, the webinar, your talk, I mean, and all this advance that is in the area of numerical relativity and everything. So I was wondering, what about these cases of massive gravity theories? So is any possible way that you can embed it in your models? And also the second question just inspired by the previous one, is what about in the case when you have this electromagnetic or neutrino emission from these mergers? Is this a way like emitting actions or a way to connect standard model physics with the gravity or the theory being that you are studying in your case? Okay, very good questions. Thank you. So the first one was about massive gravity, right? Yes. So one of these types of theories is called that I'm going to tell you, which is on the slide here. And this one, I think, is it's got better properties than some of the other ones. So the old massive gravity theories have these problems called the Bulaware Desert Ghost, which has been cured by the modern Duram Gavadatse Tolle theory. So that gives me hope that it would make more sense when you try to put it on the computer. There is a student, where is he? I don't remember where he is, but his name is Mikiča, Mikiča Kočić, something like that. I can find his name for you and point you to his papers if you're interested, who has been working on the kind of what we would call the ADM decomposition, but for bi-gravity. So bi-gravity is like a bigger version of Duram Gavadatse Tolle. Instead of just having one massive gravitan, you have two metrics and they interact with each other in the same way that is prescribed in Duram Gavadatse Tolle. But now they're both dynamical instead of one being a reference metric and one being dynamical. And this means that you have a massless gravitan that has just two polarizations and a massive gravitan that has five polarization states. And he's been working on what is necessary for that theory to have a well-posed initial value problem, just not in the way that I've been talking about from an effective field theory point of view, but just period, just does it have a well-posed initial value problem. So I'm more hopeful about bi-gravity. And there's another secret reason why I'm more hopeful about bi-gravity, which I guess is not very secret. Other people know about it. So there's a way to kind of, you know, this was done post-hoc. People derived bi-gravity and then they said, hey, wait a minute. Look, you can do this trick. What you can do is take five-dimensional general relativity, just general relativity. We know that general relativity has a well-posed initial value problem in four plus one dimensions or seven plus one dimension. It doesn't matter. And you can take one of those extra space-like dimensions and compactify it into a circle. So this is like the old Kalutsa Klein idea. And when you do this and you make certain kinds of approximations of how to treat the metric in that fifth dimension, then you can get out of it bi-gravity. You have to jump through some hoops, but the hope is that if there's some connection between five-dimensional general relativity on a compactified on a circle and bi-gravity, then there is some, you know, fully well-posed initial value problem for this five-dimensional theory that has a massive graviton when you look at it as a four-dimensional theory. So that gives me a lot of hope that massive gravity theories do have a well-posed initial or some massive gravity theories have a well-posed initial value problem. And, okay, sorry, I, sorry, I already forgot what the second question was. Now, it was about the, in the case, because when there was some merging of black holes, there were some emission in electromagnetic and neutrino observation, but they were coinciding with the, with the merging. Is there any chance that in an extended rabbit theory, that is, can be related with this kind of extra observables? Yeah, okay, that's a very good question. And actually, there were a handful of papers, like a dozen, maybe more papers, after the neutron star, neutron star merger event that used, used the gravitational wave propagation and the coincidence with the gamma ray bursts to put constraints on theories that have, say, a massive graviton or theories that are trying to mimic dark matter, actually. So some people tried to change the theory of gravity to try to make dark matter come out of the gravitational sector, which is a whole other, you know, I can talk about the philosophy of why I think that there's no real difference, but whatever. But their claim, which I actually haven't been able to track down in the literature, is that if you change the theory to, to try to mimic dark matter, then gravitational waves will propagate through galaxies at a different speed than light does. As compared to when they're outside of galaxies far where the dark matter density is very low, gravitational waves and light waves would travel at the same speed. So based on the observation of GW 170817, there were a bunch of theories that people constrained, just based on the coincidence itself. So that works under the assumption that however the gravitational waves are generated is the same when you, between general relativity and when you change the theory of gravity to whatever this other theory is. So I think that that's, that's probably okay because the portion of the waveform that LIGO saw was where the velocity over the speed of light in the system was not actually huge. I mean, it was, it was large compared to things that we're used to dealing with, but it's not so large that the V over C expansion is breaking down. So I'm okay with the idea of saying the gravitational waves that were coming out of the neutron star, neutron star merger were the same, you know, as we had in, in general relativity, and it's only the propagation between the, the source and us that might have been different. And let's use that to constrain the theory. I don't know about the other matter sector. I don't know about neutrinos. Yeah, I haven't thought about that actually. I don't know how neutrinos could help to test general relativity. Okay. Okay. And then, well, I, we have other questions and people are asking. And so we will do that. This is from Nicolas Fernandez. He said, could you comment about the nature of low values of the effective spin parameter from the binary black holes merger measure by the collaboration like Virgo. Okay. Thanks, Nicholas. Good question. Right. So you've clearly been keeping up with the literature of the 10 black hole black hole mergers that have been observed to date so far. I think nine of them, maybe eight of them have a chi effective, which is a certain combination of the spins weighted by masses that is consistent with zero. One of them was inconsistent with zero and one of them I think is maybe borderline. So I want to say two things. One is that chi effective should in principle be a tracer of where these black hole binaries formed, what their formation channel was. The idea being that if, if you have a black hole binary that formed in the field, so you had, you know, one star going supernova, then the other star evolves off of the main sequence, overflows this Roshlob and these two objects have some dynamical interactions. Those dynamical interactions, the title, the title terms between the black hole and the, you know, red giant phase star will tend to align the spins, which, which can show up in a larger value of chi effective. But if this black hole binary formed in a globular cluster, if there were exchange interactions with other compact objects in the globular cluster, that can tend to randomize the spins and that would tend to lead to lower values of chi effective. So this may be something that's intrinsic about, you know, supernovas being born, supernovas giving birth to black holes with low spins or that there are dynamical interactions in globular clusters that tend to randomize the spins. But the second thing that I wanted to say was that some of these inferences are, are very strongly dependent on the prior that you use for chi effective. So there's a paper where some combination of the authors is Katerina Chatskyonu, whose name I, I probably messed up. It's Greek. And David J. Rosa and Salvo Vitale, some combination of those authors, looked at what chi effectives you would infer, what would be your posteriors for different choices of priors for chi effective. And they found that your posteriors would be kind of different in under different choices of priors. So what that's saying is that the observations are not highly informative. If the observations were highly informative, then as long as your prior had sufficient support over all of that part of probability space, then your posterior would be highly peaked in the same part of probability of parameter space once you do the inference. But since they're in different places, that means that the observations are not sufficiently informative to say this is the value of chi effective period. It's, it's just still not the, the signal to noise ratio is not high enough. Okay. Thank you. And then the last question I promise, again, from Shantanu is are also are there models from inflation based upon Chernsheim on gravity? Yes. That's a, that's a good question. And I think the answer is yes. I remember some papers by Stefan Alexander, where it was a model that was similar to dynamical Chernsheim in the gravitational sector. But I think that it was actually coupled to a couple to SU two week or SU three or something. I don't, I don't exactly remember, but it was very similar. It was, it was in the same spirit as a Chernsheim is like theory. And it was trying to simultaneously drive inflation and also generate CP violation or something like that. So maybe that's not, you know, it's not the same dynamical Chernsheim as that we're talking about here. But it's, it's pretty similar. Okay. Thank you very much, Leo, for your time. And we also thank all of our listeners today. And then you saw Leo, and again, I'm going to show you here if you want to follow up with more questions at due to symmetry, or you can just find his web nice webpage to keep with the questions. Yeah, that's all the information. Thank you very much, Leo, and see you next in our next webinar. Hi, thank you.